EXTRACTION OF THE SQUARE ROOT 
63 
If we first found the unit 1 and then tried to find the next 
term by trial, it would probably involve a troublesome amount 
of trials. An alternative method in such a case was to 
multiply the number by 60 2 , thus reducing it to second- 
sixtieths, and then, taking the square root, to ascertain the 
number of first-sixtieths in it. Now 3.60 2 = 10800, and, as 
103 2 = 10609, the first element in the square root of 3 is 
found in this way to be ( = 1 +^). That this was the 
method in such cases is indicated by the fact that, in the Table 
of Chords, each chord is expressed as a certain number of 
first-sixtieths, followed by the second-sixtieths, &c., U3 being 
expressed as 
103 55 23 
'60 + 60 2 + 60 V 
The same thing is indicated by 
the scholiast to Eucl., Book X, who begins the operation of 
finding the square root of 31 10' 36" by reducing this to 
second-sixtieths; the number of second-sixtieths is 112236, 
which gives, as the number of first-sixtieths in the square 
335 
root, 335, while '— = 5 35'. The second-sixtieths in the 
an 
square root can then be found in the same way as in Theon’s 
example. Or, as the scholiast says, we can obtain the square 
root as far as the second-sixtieths by reducing the original 
number to fourth-sixtieths, and so on. This would no doubt 
be the way in which the approximate value 2 49' 42" 20'" 10"" 
given by the scholiast for V8 was obtained, and similarly 
with other approximations of his, such as V2 = 1 24' 51" and 
V(27) = 5 11' 46" 50'" (the 50"' should be 10'"). 
(£) Extraction of the cube root 
Our method of extracting the cube root of a number depends 
upon the formula {a + xf = a 3 + 3a 2 x + 3ax 2 + x 3 , just as the 
extraction of the square root depends on the formula 
(a + x) 2 = a 2 + 2ax + x 2 . As we have seen, the Greek method 
of extracting the square root was to use the latter (Euclidean) 
formula just as we do; but in no extant Greek writer do we 
find any description of the operation of extracting the cube 
root. It is possible that the Greeks had not much occasion 
for extracting cube roots, or that a table of cubes would 
suffice for most of their purposes. But that they had some